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Kaiden Wilkins 2022-05-07 Answered
Prove that if a , b , c R are all distinct, then a + b + c = 0 if and only if ( a , a 3 ) , ( b , b 3 ) , ( c , c 3 ) are collinear.
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Answers (1)

Ariella Bruce
Answered 2022-05-08 Author has 12 answers
Step 1
The first part of your proof is enough because you can proceed by equivalence
aligned points     equal slopes     c 2 + b c + b 2 = b 2 + a b + a 2     ( a c ) ( a + b + c ) = 0     ( a + b + c ) = 0
Here is an alternate proof:
Using the classical alignment criteria for 3 points ( x k , y k ) for k=1,2,3 which is
| x 1 x 2 x 3 y 1 y 2 y 3 1 1 1 | = 0
This means that we have to show that
| a b c a 3 b 3 c 3 1 1 1 | = 0     a + b + c = 0
But this is very easy because the determinant can be factorized in the following way:
( a c ) ( b a ) ( b c ) ( a + b + c ) ,
knowing that a,b,c are all different.
In fact, I just realized that I had already answered a similar question here... with two proofs, this one and another one based on a third degree equation with no term in x 2
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